Colorings of the Space $\mathbb R^n$ with Several Forbidden Distances

The paper is concerned with the classical problem concerning the chromatic number of a metric space, i.e., the minimal number of colors required to color all points in the space so that the distance (the value of the metric) between points of the same color does not belong to a given set of positive real numbers (the set of forbidden distances). New bounds for the chromatic number are obtained for the case in which the space is $\mathbb R^n$ with a metric generated by some norm (in particular, $l_p$) and the set of forbidden distances either is finite or forms a lacunary sequence.